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Properties

Label 18.0.11195708238...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $28.00$
Ramified primes $3, 5, 13$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![403, 1625, 3341, 4818, 4607, 3082, 1954, 1524, 1931, 1430, 445, -103, 58, 42, 0, -13, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 13*x^15 + 42*x^13 + 58*x^12 - 103*x^11 + 445*x^10 + 1430*x^9 + 1931*x^8 + 1524*x^7 + 1954*x^6 + 3082*x^5 + 4607*x^4 + 4818*x^3 + 3341*x^2 + 1625*x + 403)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 13*x^15 + 42*x^13 + 58*x^12 - 103*x^11 + 445*x^10 + 1430*x^9 + 1931*x^8 + 1524*x^7 + 1954*x^6 + 3082*x^5 + 4607*x^4 + 4818*x^3 + 3341*x^2 + 1625*x + 403, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 13 x^{15} + 42 x^{13} + 58 x^{12} - 103 x^{11} + 445 x^{10} + 1430 x^{9} + 1931 x^{8} + 1524 x^{7} + 1954 x^{6} + 3082 x^{5} + 4607 x^{4} + 4818 x^{3} + 3341 x^{2} + 1625 x + 403 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-111957082389109746826171875=-\,3^{9}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.00$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{186} a^{14} + \frac{1}{186} a^{13} - \frac{5}{186} a^{12} - \frac{5}{186} a^{11} + \frac{35}{186} a^{10} - \frac{23}{93} a^{9} - \frac{15}{62} a^{8} + \frac{4}{93} a^{7} + \frac{1}{186} a^{6} + \frac{4}{93} a^{5} - \frac{17}{186} a^{4} - \frac{1}{3} a^{3} + \frac{11}{93} a^{2} + \frac{13}{31} a - \frac{1}{2}$, $\frac{1}{2418} a^{15} + \frac{2}{1209} a^{14} + \frac{10}{403} a^{13} + \frac{4}{93} a^{12} - \frac{69}{403} a^{11} + \frac{123}{806} a^{10} - \frac{185}{806} a^{9} - \frac{437}{2418} a^{8} - \frac{37}{2418} a^{7} + \frac{13}{62} a^{6} - \frac{551}{2418} a^{5} - \frac{113}{2418} a^{4} - \frac{175}{1209} a^{3} - \frac{4}{93} a^{2} + \frac{49}{186} a - \frac{1}{6}$, $\frac{1}{5196282} a^{16} - \frac{181}{2598141} a^{15} - \frac{35}{28551} a^{14} + \frac{368521}{5196282} a^{13} + \frac{19353}{866047} a^{12} + \frac{351217}{1732094} a^{11} - \frac{704603}{2598141} a^{10} + \frac{45017}{247442} a^{9} + \frac{742790}{2598141} a^{8} - \frac{653091}{1732094} a^{7} - \frac{272639}{866047} a^{6} - \frac{184715}{742326} a^{5} + \frac{1545199}{5196282} a^{4} - \frac{1135733}{2598141} a^{3} - \frac{87142}{199857} a^{2} - \frac{4108}{9517} a + \frac{1271}{6447}$, $\frac{1}{56733801766846386} a^{17} - \frac{168877712}{9455633627807731} a^{16} - \frac{1208343717671}{28366900883423193} a^{15} - \frac{22958827703606}{28366900883423193} a^{14} - \frac{126965500884735}{18911267255615462} a^{13} + \frac{8160645908468}{2182069298724861} a^{12} - \frac{5657382317877791}{56733801766846386} a^{11} + \frac{1870081641978343}{28366900883423193} a^{10} - \frac{13943640772139254}{28366900883423193} a^{9} + \frac{2935404783541590}{9455633627807731} a^{8} - \frac{4344474378171374}{28366900883423193} a^{7} + \frac{11840292811443374}{28366900883423193} a^{6} + \frac{8828556975933083}{18911267255615462} a^{5} - \frac{86261618448619}{610040879213402} a^{4} + \frac{4344276848044710}{9455633627807731} a^{3} - \frac{323586194835266}{2182069298724861} a^{2} - \frac{225939109655828}{2182069298724861} a - \frac{41982320484565}{140778664433862}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1783988040}{176552712583} a^{17} + \frac{17500086610}{529658137749} a^{16} - \frac{30916009571}{529658137749} a^{15} + \frac{76725576665}{529658137749} a^{14} - \frac{20272690406}{529658137749} a^{13} - \frac{16999777484}{40742933673} a^{12} - \frac{264023284202}{529658137749} a^{11} + \frac{221723026506}{176552712583} a^{10} - \frac{2529834181853}{529658137749} a^{9} - \frac{2331961138812}{176552712583} a^{8} - \frac{8347298363431}{529658137749} a^{7} - \frac{1568729645500}{176552712583} a^{6} - \frac{2588681383955}{176552712583} a^{5} - \frac{440136406252}{17085746379} a^{4} - \frac{20545061088497}{529658137749} a^{3} - \frac{1349018629990}{40742933673} a^{2} - \frac{769741752119}{40742933673} a - \frac{7273288033}{1314288183} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 613727.005399 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.12675.1 x3, 3.3.169.1, 6.0.481966875.1, 6.0.2851875.1 x2, 6.0.771147.1, 9.3.2036310046875.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.2851875.1
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3_13.6t1.1c1$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3_13.6t1.1c2$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.3_5e2_13e2.3t2.1c1$2$ $ 3 \cdot 5^{2} \cdot 13^{2}$ $x^{3} - x^{2} + 22 x + 12$ $S_3$ (as 3T2) $1$ $0$
*2 2.3_5e2_13.6t5.3c1$2$ $ 3 \cdot 5^{2} \cdot 13 $ $x^{18} - 3 x^{17} + 5 x^{16} - 13 x^{15} + 42 x^{13} + 58 x^{12} - 103 x^{11} + 445 x^{10} + 1430 x^{9} + 1931 x^{8} + 1524 x^{7} + 1954 x^{6} + 3082 x^{5} + 4607 x^{4} + 4818 x^{3} + 3341 x^{2} + 1625 x + 403$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3_5e2_13.6t5.3c2$2$ $ 3 \cdot 5^{2} \cdot 13 $ $x^{18} - 3 x^{17} + 5 x^{16} - 13 x^{15} + 42 x^{13} + 58 x^{12} - 103 x^{11} + 445 x^{10} + 1430 x^{9} + 1931 x^{8} + 1524 x^{7} + 1954 x^{6} + 3082 x^{5} + 4607 x^{4} + 4818 x^{3} + 3341 x^{2} + 1625 x + 403$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.